12 HIGHER DERIVATIVES 185

Show that the function p is indefinitely differentiable in R. (Use the relation
lim ^e-x

(b) In this problem, we agree to extend any regulated function /defined in a compact
interval [a, b] of R, to the whole of R, by giving it the value 0 for t < a and for t > b;

/• + oo /*b /ğd

we then write f(t) dt for the integral f(t) dt, which is also equal to /(/) dt

J-oo Ja Jc

for c ^ a and d^s-b.
For any such function/, let

fn(t) = ncr "VfcW ™ *)) ds-ncT °/0 ~ s)p(ns) ds

f+1
where l/c= p(f) dt (*' regularization " of / by p; we write pn(t) — ncp(nt) and

fn=f*pn). Show that /„ is indefinitely differentiate and vanishes in the comple-
ment of a compact interval (use (8.11.2); if/ is real and increasing (resp. strictly
increasing, resp. convex) in [a, b], then /„ is increasing (resp. strictly increasing,
resp. convex) in [a + l//i, 6 — l//i]. If /(extended to R) is;? times continuously differ-
entiable, then

f +

J-OO

- a)) &

/. + OD /ğ+co

(c) Show that for any n, fn(t) dt = f(t) dt.

J-oo J-oo

(d) If / (extended to R) is continuous (resp. p times continuously differentiable),
then the sequence (/„) (resp. Dp/n) converges uniformly in R to /(resp. Dp/).

(e) To what limit does/,(/o) (*o e R) tend when /is only supposed to be regulated in
[a, b] (first consider the case in which /is a step-function, then use (7.6.1)).

(f) Show that for any regulated function /in [a, b],

lim

n-too

3. Let /be an n times differentiable real function defined in ]—!,![ and such that
|/(/)| ^ 1 in that interval.

(a) Let 772*0) be the smallest value of \f(k\t)\ in an interval J contained in ]— 1, 1[.
Show that, if J is decomposed into three consecutive intervals Ji, J2, Ja, and if J2
has length p,, then, for k < n,

(use the mean value theorem). Deduce from that inequality that if J has length A,

- 5T—

A

(use induction on k).

(b) Deduce from (a) that there exists a number ocn depending only on n, such that if
l/'(0)| ^ an,/(ll)(0 = 0 has at least n — 1 distinct roots in ]— 1, 1[. (Show by induction
on k that there is a strictly increasing sequence xkti<xkt2< "' < xktk of points
of ]-l, 1[ such that fck\xk, i)f(k\xk, t+i) < 0 for 1 =s? / ^ k - I ; use Rolle's theorem.)21).